Tidal dissipation in a homogeneous spherical body. I. Methods

TL;DR

This paper derives a corrected formula for tidal dissipation in homogeneous spherical bodies, improving upon previous models by ensuring positive-definite contributions from all Fourier modes, with implications for planetary tidal studies.

Contribution

It introduces a mathematically accurate and comprehensive formula for tidal dissipation, correcting inaccuracies in prior literature and integrating the Darwin-Kaula formalism.

Findings

Derived a new, mathematically correct tidal dissipation formula.

Compared with previous models, showing differences in Fourier mode contributions.

Validated the formula with applications to planetary tidal damping.

Abstract

A formula for the tidal dissipation rate in a spherical body is derived from first principles, to correct some mathematical inaccuracies found in the literature. The development is combined with the Darwin-Kaula formalism for tides. Our intermediate results are compared with those by Zschau (1978) and Platzman (1984). When restricted to the special case of an incompressible spherical planet spinning synchronously without libration, our final formula can be compared with the commonly used expression from Peale & Cassen (1978, Eqn. 31). The two turn out to differ. In our expression, the contributions from all Fourier modes are positive-definite, this not being the case of the formula from Ibid. (The presence of negative terms in their formula was noticed by Makarov 2013.) Examples of application of our expression for the tidal damping rate are provided in the work by Makarov and Efroimsky (2014).